We are given a polynomial function
in factored form and asked to determine the zeros
and multiplicity, as well as the degree and end behavior.
The zeroes of the polynomial function
are the x values where the function value
is zero, and because the polynomial function's
in factored form, the values of x
that make each factor equal to zero
are the zeroes of the polynomial.
So, looking at the first factor of x plus one,
we know x plus one is equal to zero
when x equals negative one.
X equals negative one is a zero of the polynomial function,
but notice how we have the quantity x plus one
raised to the fourth, so because we have four factors of x
plus one, the zero of negative one has multiplicity of four.
Next, the factor of x plus two is equal to zero
when x equals negative two.
Negative two is a zero of the polynomial function,
but because of the exponent of two,
we know we have two factors of x plus two, and therefore
the zero of negative two has a multiplicity of two.
Next, the factor of five x plus one is zero.
One x equals negative 1/5.
If we can't tell that by inspection, we can always
set the factor of five x plus one equal to zero and solve.
The first step would be to subtract one on both sides,
giving us five x equals negative one,
and then divide both sides by five,
and we do get x equals negative 1/5,
which is a zero of the polynomial function,
but because the factor of five x plus one
is raised to the second power,
this zero has multiplicity two.
And then, finally, the factor of x minus two
is equal to zero when x equals positive two.
x equals positive two is a zero
of the polynomial function, but because
the factor of x minus two is raised to the third power,
this zero of x equals two has multiplicity three.
Next, we're asked to determine the degree
of the polynomial function.
The degree of the polynomial function is the same
as the degree of the term with the highest degree.
Let's analyze this function and see
if we can find the leading term,
or the term that would have the highest degree.
So, at the beginning, we know we have a factor
of three, then the term with the highest degree
from the quantity x plus one raised to the fourth
would be x to the fourth, so multiply
by x to the fourth, times the term with the highest degree
from the quantity x plus two squared would be x squared,
times the term with the highest degree
from the quantity five x plus one squared
would be the square of five x, or 25 x squared,
and then, finally, the term with the highest degree
from the quantity x minus two cubed would be x cubed.
So now, multiplying, three times 25 is 75,
and then we have x to the fourth times x to the second,
times x to the second, times x to the third.
When multiplying, and the bases are the same,
we add the exponents.
Four plus two plus two plus three is 11,
giving us x to the 11th.
So, this would be the leading term
of the polynomial function, or the term
with the highest degree, and notice
how the degree of this term is 11,
which means the degree of the polynomial is also 11.
And now, for the last part, we're asked
to determine the end behavior of the polynomial function.
The end behavior of the given polynomial function
would be the same end behavior as the power function
g of x equals 75 x raised to the power of 11,
which means we should be able to determine
the end behavior by knowing the leading coefficient
is positive, because it's 75, and the degree
is odd, because the exponent, 11, is odd.
If we go back to our notes, as long as we know
the leading term, or the term with the highest degree,
of any polynomial, we can determine the end behavior
just by using that first term,
where if the term is in the form a times x
raised to the power of n, if a is positive,
then we can find the end behavior in this row.
If a is negative, we can find the end behavior in this row.
So, in our case, we know a is positive,
and the degree is 11, which is odd, and therefore,
this gives us end behavior of our polynomial function.
Whereas x approaches negative infinity,
or as you move to the right, the function
values or y values going down without bound,
approaching negative infinity, and as x approaches
positive infinity, or you move to the right,
the function values or y values
increase without bound, and approach positive infinity.
Remember, function values are along the vertical axis.
Or, of course, if we needed to,
we could always graph the original function,
or we could just graph the first term
of the polynomial function, which is 75 x to the 11th.
Let's take a look at both of these graphs.
So, here's the graph of the original function,
which might take some time to graph
to adjust the window properly,
and here's a graph of g of x equals 75 x to the 11th,
which would be the first term, or the term
of the highest degree of the original polynomial function.
So, notice, as x approaches negative infinity,
or as we move to the left along the graph,
in this direction, notice how the y values,
or function values, are decreasing without bound
and are approaching negative infinity, and the same thing
is true for g of x equals 75 x to the 11th.
As x approaches negative infinity,
or as they move to the left, the function values
or y values decrease without bound,
approaching at negative infinity.
This does verify, as x approaches negative infinity,
f of x approaches negative infinity,
and then as x approaches positive infinity,
along the given function, we're moving to the right.
Notice as we move to the right,
the function goes up, without bound,
meaning f of x, or y, approaches positive infinity,
and the same thing for g of x equals 75 x to the 11th.
As we move to the right, or as x approaches infinity,
f of x, or y, increases without bound
and approaches positive infinity.
So, as x approaches infinity,
f of x also approaches infinity.
I hope you found this helpful.